Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 25, pp. 1–15. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.25 PERIODIC SOLUTIONS IN DISTRIBUTION FOR STOCHASTIC LATTICE DIFFERENTIAL EQUATIONS YUE GAO, XUE YANG Abstract. In this article, we consider stochastic lattice differential equations (SLDEs) in the weighted space l2ρ of infinite sequences. We establish the well- posedness of solutions and prove the existence of periodic solutions in distri- bution. An example is given to illustrate the validity of our results. 1. Introduction Lattice differential equations have been extensively studied because the variety of applications in image processing, traffic flow analysis, virus propagation, pattern formation, and so on. For the dynamics of deterministic lattice differential equa- tions, we refer the reader to [2, 6, 13, 24] and references therein. Compared to deterministic systems, stochastic lattice systems not only exhibit discrete spatial characteristics but also account for the influence of random environments. This en- ables SLDEs to better reveal objective phenomena. For this reasing, SLDEs have attracted extensive attention; see [3, 4, 8, 14, 15, 26]. The concept of periodic solutions plays a crucial role in studying the long-term behavior of random dynamical systems simulated by stochastic differential equa- tions. Since the ground breaking work of Poincaré in [21, 22, 23], periodic solutions have been the subject of research for over a century. In the past decade, many works have been devoted to study periodicity of SDEs. For the existence of pe- riodic solutions for finite-dimensional stochastic systems, we refer the reader to [5, 9, 10, 11, 12, 16, 17, 18, 28]. Similar to the case of finite-dimensional systems, a crucial question is: Under what conditions do SLDEs in weighted space l2ρ have the desired periodicity? In this article, we focus on asymptotic behavior and attempt to address this issue. Despite the increasing interest in treating SLDEs, the available results in this re- gard still scarce. There are two main difficulties. First, because the disturbance from noise, the sample paths of the solutions cannot maintain periodicity. In ad- dition, rigorous convergence analysis is required to ensure the well-posedness of solutions of infinite-dimensional systems. To this end, we consider a weaker period- icity, so-called periodic solutions in distribution. In this paper, we first discuss the well-posedness of SLDEs. Inspired by [5, 12], we provide sufficient conditions for 2020 Mathematics Subject Classification. 34C25, 34C27, 37H10. Key words and phrases. Periodic solutions; stochastic lattice differential equations; weighted spaces. ©2024. This work is licensed under a CC BY 4.0 license. Submitted September 9, 2023. Published March 21, 2024. 1 2 Y. GAO, X. YANG EJDE-2024/25 the existence of periodic solutions in distribution for general SLDSs in a weighted space of infinite sequences. Furthermore, we provide an illustrative example to demonstrate the simplicity of our conditions via Lyapunov method. The rest of this article is organized as follows. In Section 2, we give some preliminaries. We introduce the notation and definitions of related concepts. In addition, we discuss the well-posedness of SLDEs. In Section 3, we first give a priori estimate to ensure the rationality of the assumptions. Then, we prove the existence of periodic solutions in distribution. In Section 4, we illustrate our main result by an example. 2. Preliminaries Basic notation. First, we introduce a weighted space of infinite sequences. Let ρ : Z → (0,M0] ⊂ R+ and p ≥ 1 be a real number. For each i ∈ Z, we define ρ(i) = ρi and lpρ = { u = (ui)i∈Z; ∞∑ i=1 ρi|ui|p < ∞ } with the norm ∥u∥ρ,p = (∑∞ i=1 ρi|ui|p )1/p for u ∈ lpρ. If p = 2, we denote ∥u∥ρ,2 = ∥u∥ρ. For u, v ∈ l2ρ, we denote the inner product in l2ρ as ⟨u, v⟩, where ⟨u, v⟩ =∑ i∈Z ρiuivi. The space Lp(Ω, l2ρ) consists of all l 2 ρ-valued random variables ξ such that E∥ξ∥pρ = ∫ Ω ∥ξ∥pρdP < ∞. For a given l2ρ-valued random variable ξ, we denote by P ◦ [ξ]−1 the distribution of ξ on l2ρ. Let B(l2ρ) be the Borel set of space l2ρ. For z ∈ l2ρ, we use zT to denote the transpose of z. Let P(l2ρ) be the set of Borel probability measures on l2ρ. Denote by Jf the Jacobian matrix of function f with respect to x ∈ Rd. We define ∥h∥∞ = sup x∈l2ρ |h(x)|, ∥h∥L = sup { |h(x)− h(y)| ∥x− y∥ρ ;x, y ∈ l2ρ, x ̸= y } , ∥h∥BL = max{∥h∥∞, ∥h∥L}, dBL(µ1, µ2) = sup ∥h∥BL≤1 ∣∣ ∫ hd(µ1 − µ2) ∣∣ for all Lipschitz continuous real-valued functions h(x) on l2ρ and all µ1, µ2 ∈ P(l2ρ). Well-posedness of SLDEs. Let (Ω,F , {Ft}t≥0, P ) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is increas- ing, right continuous and F0 contains all P -null sets). The first component u(t) satisfies the SLDE dui(t) = [ν(ui+1(t)− 2ui(t) + ui−1(t))− λui(t) + fi(u(t)) + gi(t)]dt + σi(t, u(t))dWi(t), (2.1) where i ∈ Z, ui ∈ R, ν, and λ are positive constants. We assume that (fi)i∈Z are smooth functions, (gi(t))i∈Z, (σi(t))i∈Z ∈ l2ρ are continuous with respect to t ∈ R+, and {Wi(t) : i ∈ Z} are independent one-dimensional Brownian motions. For u ∈ l2ρ, let A, B, and B∗ be linear operators from l2ρ to l2ρ as follows: (Bu)i = ui+1 − ui, (B∗u)i = ui−1 − ui, EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 3 and (Au)i = −ui+1 + 2ui − ui−1. Then we have A = BB∗ = B∗B and ⟨B∗u, v⟩ = ⟨u,Bv⟩ for all u, v ∈ l2ρ. Therefore, ⟨Au, u⟩ ≥ 0 for all u ∈ l2ρ. Let ei denote the element having 1 at position i and all the other components 0. We define W (t) = ∑ i∈Z Wi(t)e i, f(u(t)) = (fi(u(t)))i∈Z, g(t) = (gi(t))i∈Z, σ(t, u(t)) = (σ̃ij(t, u(t)))i,j∈Z, where σ̃ij = { σi, i = j, 0, i ̸= j. Then we can rewrite (2.1) as du(t) = [−νAu(t)− λu(t) + f(u(t)) + g(t)]dt+ σ(t, u(t))dW (t). (2.2) Note that (2.2) can be interpreted as an integral equation u(t) = u0 + ∫ t 0 [−νAu(s)− λu(s) + f(u(s)) + g(s)]ds+ ∫ t 0 σ(s, u(s))dW (s) (2.3) with initial value u0 := u(0). We make the following assumptions on the coefficients of the above SLDE. Assumption 2.1. For every i ∈ Z, t ∈ [0,∞), and u, v ∈ l2ρ, there exists positive constants L and K such that ∥f(u)− f(v)∥2ρ ≤ L∥u− v∥2ρ, ∥f(u)∥2ρ ≤ K(1 + ∥u∥2ρ), ∥σ(t, u)− σ(t, v)∥2ρ ≤ L∥u− v∥2ρ, ∥σ(t, u)∥2ρ ≤ K(1 + ∥u∥2ρ). Next, we prove the existence and uniqueness of solutions to SLDEs. Theorem 2.2. Let T > 0, and suppose that Assumptions 2.1 holds. Then (2.3) admits a unique solution u(t) ∈ L2(Ω, C([0, T ], l2ρ)) with initial value u(0) = u0 ∈ L2(Ω, l2ρ). Proof. Step 1. We show uniqueness. Assume that u(t) and ũ(t) are two solution of system (2.3) with initial value u0 ∈ l2ρ. Then we have u(t)− ũ(t) = ∫ t 0 [−νA(u(s)− ũ(s))− λu(s) + λũ(s)) + f(u(s))− f(ũ(s))]ds+ ∫ t 0 [σ(s, u(s))− σ(s, ũ(s))]dW (s). Hence, by Itô isometry and Assumption 2.1, we have E∥u(t)− ũ(t)∥2ρ = E∥ ∫ t 0 [−νA(u(s)− ũ(s))− λ(u(s)− ũ(s)) + f(u(s))− f(ũ(s))]ds + ∫ t 0 [σ(s, u(s))− σ(s, ũ(s))]dW (s)∥2ρ ≤ 2tE ∫ t 0 ∥ − νA(u(s)− ũ(s))− λ(u(s)− ũ(s)) + f(u(s))− f(ũ(s))∥2ρds + 2E ∫ t 0 ∥σ(s, u(s))− σ(s, ũ(s))∥2ρds 4 Y. GAO, X. YANG EJDE-2024/25 ≤ 6tE ∫ t 0 ∥ − νA(u(s)− ũ(s))∥2ρds+ 6tE ∫ t 0 λ2∥u(s)− ũ(s)∥2ρds + 6tE ∫ t 0 ∥f(u(s))− f(ũ(s))∥2ρds+ 2E ∫ t 0 ∥σ(s, u(s))− σ(s, ũ(s))∥2ρds. Note that ∥νA(u(s)− ũ(s))∥2ρ = ∑ i∈Z ρi ( ν2 ∑ i∈Z [(ui+1(s)− ũi+1(s))− 2(ui(s)− ũi(s)) + (ui−1(s)− ũi−1(s))] 2 ) ≤ 18ν2∥u(s)− ũ(s)∥2ρ. So, E∥u(t)− ũ(t)∥2ρ ≤ [6t(18ν2 + λ2 + L) + 2L]E ∫ t∧ξ 0 ∥u(s)− ũ(s)∥2ρds. By Grownwall’s inequality, we obtain E∥u(t)− ũ(t)∥2ρ = 0. This means that P{u(t) = ũ(t)} = 1 for all t ≥ 0. Step 2. We claim that (2.3) admits a solution. Let u0(t) = u0. For each n = 1, 2, . . . , we define the Picard iterations un(t) = u0 + ∫ t 0 [−νAun−1(s)− λun−1(s) + f(un−1(s)) + g(s)]ds + ∫ t 0 σ(s, un−1(s))dW (s). (2.4) Hence E[∥un(t)− un−1(t)∥2ρ] = E[∥ ∫ t 0 [−νAun−1(s)− λun−1(s) + f(un−1(s)) + g(s) − (−νAun−2(s)− λun−2(s) + f(un−2(s)) + g(s))]ds + ∫ t 0 [σ(s, un−1(s))− σ(s, un−2(s))]dW (s)∥2ρ] ≤ 2tE[ ∫ t 0 ∥ − νA(un−1(s)− un−2(s))− λ(un−1(s)− un−2(s)) + f(un−1(s))− f(un−2(s)))∥2ρds] + 2E ∫ t 0 ∥σ(s, un−1(s))− σ(s, un−2(s))∥2ρds ≤ 6tE[ ∫ t 0 ∥ − νA(un−1(s)− un−2(s))∥2ρ + ∥ − λ(un−1(s)− un−2(s))∥2ρ + ∥f(un−1(s))− f(un−2(s))∥2ρds] + 2E ∫ t 0 ∥σ(s, un−1(s))− σ(s, un−2(s))∥2ρds ≤ [6t(18ν2 + λ2 + L) + 2L]E ∫ t 0 ∥un−1(s)− un−2(s)∥2ρds. In addition, E[∥u1(t)− u0(t)∥2ρ] EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 5 = E[∥ ∫ t 0 [−νAu0(s)− λu0(s) + f(u0(s)) + g(s)]ds+ ∫ t 0 σ(s, u0(s))dW (s)∥2ρ] ≤ 8tE ∫ t 0 [∥ − νAu0(s)∥2ρ + ∥λu0(s)∥2ρ + ∥f(u0(s))∥2ρ + ∥g(s)∥2ρ]ds + 2E ∫ t 0 ∥σ(s, u0(s))∥2ρds ≤ 8tE ∫ t 0 [18ν2∥u0(s)∥2ρ]ds+ 8tE ∫ t 0 λ2∥u0(s)∥2ρds+ 8tE ∫ t 0 K(1 + ∥u0(s)∥2ρ)ds + 8K1t 2 + 2E ∫ t 0 K(1 + ∥u0(s)∥2ρ)ds ≤ [8t2(18ν2 + λ2 +K) + 2Kt]E∥u0(s)∥2ρ + 8(K +K1)t 2 + 2Kt, where K1 = maxs∈[0,T ] ∥g(s)∥2ρ. Then there exists a positive constant C1 < ∞ such that E∥u1(t)− u0(t)∥2ρ ≤ C1t, where C1 only depends on ν, λ,K,K1, T . By induction, there exists a positive constant C2 such that for any n ≥ 0, t ∈ [0, T ], we have E∥un(t)− un−1(t)∥2ρ ≤ Cn 2 t n n! , where C2 only depends on ν, λ, K, K1, T, L and C2 ≥ max{C1, 6T (18ν 2 + λ2 + L) + 2L}. In addition, E ( sup 0≤t≤T ∥un(t)− un−1(t)∥2ρ ) ≤ [6T (18ν2 + λ2 + L) + 2L]E ∫ T 0 ∥un−1(s)− un−2(s)∥2ρds ≤ C2 ∫ T 0 Cn−1 2 sn−1 (n− 1)! ds = C2T n n! . By Chebyshev’s inequality, we obtain P { sup 0≤t≤T ∥un(t)− un−1(t)∥ ≥ 1 2n } ≤ (4C2T ) n n! . Note that ∑∞ n=1 (4C2T )n n! < ∞. Hence by Borel-Cantelli’s lemma, for almost all ω ∈ Ω, there exists an integer constant n0 such that sup 0≤t≤T ∥un(t)− un−1(t)∥2ρ ≤ 1 2n for n ≥ n0. Consequently, un(t) converges to u(t) as n → ∞ uniformly in t ∈ [0, T ] for almost all ω. Note also that E∥un(t)∥2ρ = E∥u0 + ∫ t 0 [−νAun−1(s)− λun−1(s) + f(un−1(s)) + g(s)]ds + ∫ t 0 σ(s, un−1(s))dW (s)∥2ρ 6 Y. GAO, X. YANG EJDE-2024/25 ≤ 3E∥u0∥2ρ + [12T (18ν2 + λ2 +K) + 3K]E ∫ t 0 ∥un−1(s)∥2ρds + 12KT 2 + 12K1T 2 + 3KT. From this inequality, for all k ≥ 1, we have max 1≤n≤k E∥un(t)∥2ρ ≤ 3E∥u0∥2ρ + [12T (18ν2 + λ2 +K) + 3K]E ∫ t 0 max 1≤n≤k ∥un−1∥2ρds + 12KT 2 + 12K1T 2 + 3KT ≤ 3E∥u0∥2ρ + [12T (18ν2 + λ2 +K) + 3K]E ∫ t 0 [∥u0∥2ρ + max 1≤n≤k ∥un∥2ρ]ds + 12KT 2 + 12K1T 2 + 3KT ≤ 3E∥u0∥2ρ + 12KT 2 + 12K1T 2 + 3KT + [12T (18ν2 + λ2 +K) + 3KT ]E∥u0∥2ρ + [12T (18ν2 + λ2 +K) + 3K] ∫ t 0 [ max 1≤n≤k E∥un∥2ρ]ds. Let K3 = 12KT 2+12K1T 2+3KT +[12T (18ν2+λ2+K)+3+3KT ]E∥u0∥2ρ. Then by Gronwall’s inequality, it holds that max 1≤n≤k E∥un(t)∥2ρ ≤ K3e 12T 2(18ν2+λ2+K)+3KT . So, we have E∥un(t)∥2ρ ≤ K3e 12T 2(18ν2+λ2+K)+3KT , for t ∈ [0, T ], n ≥ 1, which implies that E∥u(t)∥2ρ < ∞ for t ∈ [0, T ]. We proceed to prove that u(t) satisfies system (2.4) with u0 ∈ l2ρ × S. It is not difficult to verify that E[∥ ∫ t 0 [−νA(un(s)− u(s))− λ(un(s)− u(s)) + (f(un(s))− f(u(s)))]ds + ∫ t 0 σ(s, un(s))− σ(s, u(s))dW (s)∥2ρ] ≤ [6T (18ν2 + λ2 + L) + 2L]E ∫ t 0 ∥un(s)− u(s)∥2ρds → 0, as n → ∞. Hence u(t) satisfies (2.3). □ The proof of Theorem 2.2 is inspired by proofs of [19, Theorem 3.1] and [27, Theorem 3.1]. 3. Existence of periodic solutions In this section, we establish the criterion for the existence of the periodic solution in distribution of (2.3) in l2ρ. First, we give the definition for the periodic solution in distribution. Definition 3.1. A solution u(t) of (2.3) is said to be a θ-periodic solution in distribution if for any t ∈ R+, u(t) satisfies the following conditions: (i) P ◦ [u(t)]−1 = P ◦ [(u(t+ θ))]−1; EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 7 (ii) there exists W (t) such that u(t+ θ) is a solution of the equation du(t) = [−νAu(t)− λu(t) + f(u(t)) + g(t)]dt+ σ(t, u)dW (t), where W (t) has the same distribution with W (t). Definition 3.2. A sequence of probability measures µn ∈ P(l2ρ) is said to be weakly convergent to a probability measure µ ∈ P(l2ρ), if∫ l2ρ f(x)µn(dx) → ∫ l2ρ f(x)µ(dx) as n → ∞, where f(x) is any continuous bounded function on l2ρ. Definition 3.3. A sequence of l2ρ-valued stochastic processes {Xn(t)} is said to be convergent in distribution to an l2ρ-valued stochastic process X(t) if the distribution of {Xn(t)} converges weakly to the distribution of X(t) for all t ∈ R+. Next, we estimate the p-th moment of the solution u(t). Lemma 3.4. Let p > 2 and ξ ∈ Lp(Ω, l2ρ). Suppose that Assumption 2.1 holds. Then for all t ∈ [0, T ], E( sup 0≤s≤t ∥u(t)∥pρ) ≤ (1 + 3p−1E∥u0∥pρ)eat, where a = max { (12T )p−1[3p−1(2 + 2p)νp + λp + 2 p 2−1Kp/2] + 3p−12 p 2−1 (p(p− 1) 2 )p/2 T p−2 2 , (2 p 2−1Kp/2 +K1)(12T ) p−1 + 3p−12 p 2−1 (p(p− 1) 2 )p/2 T p−2 2 } . Proof. Note that ∥ − νAu(s)∥pρ = ∑ i∈Z ρi[ν p(ui+1(s)− 2ui(s) + ui−1(s)) p] ≤ 3p−1νp ∑ i∈Z [|ui+1(s)|p + 2p|ui(s)|p + |ui−1(s)|p] ≤ 3p−1(2 + 2p)νp∥u(s)∥pρ. (3.1) By Hölder inequality, [20, Theorem 1.7.2], Assumption 2.1, and (3.1), we obtain E ( sup 0≤s≤t ∥u(t)∥pρ ) = E ( sup 0≤s≤t ∥u0 + ∫ s 0 [−νAu(s)− λu(s) + f(u(s)) + g(r)]dr + ∫ s 0 σ(r, u(r))dW (r)∥pρ ) ≤ 3p−1E∥u0∥pρ + (12t)p−1E[ ∫ t 0 ∥ − νAu(s)− λu(s) + f(u(s)) + g(s)∥pρds] + 3p−1E ( sup 0≤s≤t ∥ ∫ s 0 σ(r, u(r))dW (r)∥pρ ) 8 Y. GAO, X. YANG EJDE-2024/25 ≤ 3p−1E∥u0∥pρ + (12t)p−1[E ∫ t 0 ∥ − νAu(s)∥pρds+ E ∫ t 0 ∥ − λu(s)∥pρds + E ∫ t 0 ∥f(u(s))∥pρds+ E ∫ t 0 ∥g(s)∥pρds] + 3p−1 (p(p− 1) 2 )p/2 T p−2 2 3p−1E ∫ t 0 ∥σ(s, u(s))∥pρds ≤ 3p−1E∥u0∥pρ + (12T )p−1[3p−1(2 + 2p)νp + λp + 2 p 2−1Kp/2]E ∫ t 0 ∥u(t)∥pρds + (12T )p−12 p 2−1Kp/2t+ (12T )p−1K1t + 3p−12 p 2−1 (p(p− 1) 2 )p/2 T p−2 2 Kp/2E ∫ t 0 (1 + E∥u(s)∥2ρ)ds ≤ 3p−1E∥u0∥pρ + a ∫ t 0 (1 + E∥u(s)∥pρ)ds, where a = max { (12T )p−1[3p−1(2 + 2p)νp + λp + 2 p 2−1Kp/2] + 3p−12 p 2−1 (p(p− 1) 2 )p/2 T p−2 2 , (2 p 2−1Kp/2 +K1)(12T ) p−1 + 3p−12 p 2−1 (p(p− 1) 2 )p/2 T p−2 2 } . Hence 1 + E ( sup 0≤s≤t ∥u(t)∥pρ ) ≤ 1 + 3p−1E∥u0∥pρ + a ∫ t 0 [ 1 + E ( sup 0≤r≤s ∥u(r)∥pρ )] ds. It follows from Gronwall’s inequality that 1 + E ( sup 0≤s≤t ∥u(t)∥pρ ) ≤ (1 + 3p−1E∥u0∥pρ)eat for t ∈ [0, T ]. Therefore, we obtain E ( sup 0≤s≤t ∥u(t)∥pρ ) ≤ (1 + 3p−1E∥u0∥pρ)eat for t ∈ [0, T ]. □ For (2.3), we make the following assumptions. Assumption 3.5. Suppose that all the time-dependent coefficient functions are θ-periodic in t ∈ R+; that is, for all t ∈ R+, i ∈ Z, u ∈ l2ρ gi(t+ θ) = gi(t), σi(t+ θ, u) = σi(t, u). Assumption 3.6. For some p > 2 and n = 0, 1, 2, . . . , there exists a positive constant C independent of n such that E∥u(nθ)∥pρ ≤ C. Assumption 3.7. The distribution P ◦ [u(t)]−1 with respect to u(t) satisfies lim k→∞ 1 nk + 1 nk∑ m=0 dBL(P ◦ [u((m+ 1)θ)]−1, P ◦ [u(mθ)]−1) = 0, EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 9 where {nk} is a sequence of integers tending to +∞. Lemma 3.4 ensures that Assumption 3.6 is reasonable. In Section 4, we give an example to verify Assumptions 3.6 and 3.7. Definition 3.8. A family of random variables H in L1(Ω, l2ρ) is uniformly inte- grable, if it satisfies sup ξ∈H ∫ ∥ξ∥ρ≥M ∥ξ∥ρdP → 0 as M → ∞. Theorem 3.9. Suppose that Assumptions 2.1-3.7 hold. Then there exists a θ- periodic solution in distribution of (2.3). Proof. Let γk be a random variable independent of W (t) and u(0, ω) such that P{γk = Nθ} = 1 k + 1 , N = 0, 1, . . . , k, for each k ∈ Z+. We define a sequence of stochastic processes vk(t) = u(t+ γk), vk(0) = u(γk). Then vk(t) is a weak solution of (2.3). In fact, for C ∈ B(l2ρ) and t ∈ R+, we define W (t) = W (t+ γk)−W (γk), where W (t) has the same distribution with W (t). Hence we obtain u(t+ γk) = u(0) + ∫ γk 0 [−νAu(s)− λu(s) + f(u(s)) + g(s)]ds+ ∫ γk 0 σ(s, u(s))dW (s) + ∫ t+γk γk [−νAu(s)− λu(s) + f(u(s)) + g(s)]ds+ ∫ t+γk γk σ(s, u(s))dW (s) = u(γk) + ∫ t+γk γk [−νAu(s)− λu(s) + f(u(s)) + g(s)]ds+ ∫ t+γk γk σ(s, u(s))dW (s) = u(γk) + ∫ t 0 [−νAu(s+ γk)− λu(s+ γk) + f(u(s+ γk)) + g(s+ γk)]ds+ ∫ t 0 σ(s+ γk, u(s+ γk))dW (s). From the construction of vk(t) and the independence of γk, we have P{vk(t) ∈ A} = P{u(t+ γk) ∈ A} = P{u(t+ γk) ∈ A|γk = 0}P{γk = 0} + P{u(t+ γk) ∈ A|γk = θ}P{γk = θ}+ . . . + P{u(t+ γk) ∈ A|γk = kθ}P{γk = kθ} = 1 k + 1 k∑ N=0 P{u(t+Nθ) ∈ A} (3.2) 10 Y. GAO, X. YANG EJDE-2024/25 for each A ∈ B(l2ρ). From (3.2), Assumption 3.6, and Chebyshev’s inequality, it follows that uniformly in k, P{∥vk(0, ω)∥ρ > R} = 1 k + 1 k∑ N=0 P{∥u(Nθ)∥ρ > R} ≤ 1 k + 1 k∑ N=0 E∥u(Nθ)∥2ρ R2 → 0 as R → ∞. So, vk(0, ω) satisfies conditions of [7, Theorem 7.2]. According to Skorohod theorem [25, p 13] in another probability space (Ω̃, F̃ , P̃ ), there exists a sequence ṽk(0, ω̃) (k = 0, 1, . . . ) with the same distribution as vk(0, ω). Furthermore, there exists a subsequence ṽnk (0, ω̃) that converges to ṽ(0, ω̃) in probability. We can construct l2ρ-valued random variables v(0, ω) and vnk (0, ω) on (Ω,F , P ) with the same distri- bution as ṽ(0, ω̃) and ṽnk (0, ω̃), respectively. From Assumption 3.6, we have E∥ṽnk (0, ω̃)∥pρ = E∥vnk (0, ω)∥pρ ≤ C < ∞. for some p > 2. By [1, Proposition 2.5.7], ∥ṽnk (0, ω̃)∥2ρ is uniformly integrable. It follows from [1, Theorem 2.5.9] that for every ε > 0, there exists a δ > 0 such that for any A ∈ F with P (A) ≤ δ, we have supξ∈H ∫ A ∥ṽnk (0, ω̃)∥2ρdP ≤ ε. According to Vitali’s convergence theorem, we have E∥ṽnk (0, ω̃)− ṽ(0, ω̃)∥2ρ → 0 as nk → ∞. Let ṽnk (t) be the solution of the equation du(t) = [−νAu(t)− λu(t) + f(u(t)) + g(t)]dt+ σ(t, u(t))dW (t), with initial condition ṽnk (0, ω̃) = ṽnk (ω̃) on the probability space (Ω̃, F̃ , P̃ ). By Cauchy-Schwarz’s inequality, Itô’s isometry, and Assumption 2.1, we obtain E∥ṽnk (t)− ṽ(t)∥2ρ ≤ 3E∥ṽnk (0, ω̃)− ṽ(0, ω̃)∥2ρ + [9t(18ν2 + λ2 + L) + 3L]E ∫ t 0 ∥ṽnk (s)− ṽ(s)∥2ρds. By Gronwall’s inequality, we have E∥ṽnk (t)− ṽ(t)∥2ρ ≤ 3E∥ṽnk (0, ω̃)− ṽ(0, ω̃)∥2ρe9t 2(18ν2+λ2+L)+3Lt → 0 as nk → ∞. It follows from the uniqueness of weak solution that P ◦ [vnk (t)]−1 = P ◦ [ṽnk (t)]−1 → P ◦ [ṽ(t)]−1 (3.3) uniformly on [0, θ]. In addition, v(0, ω) on (Ω,F , P ) has the same distribution as ṽ(0, ω) on (Ω̃, F̃ , P̃ ). From the uniqueness of the weak solution of (2.3), v(t) admits the same distribution with ṽ(t). By (3.3), (3.2) and Assumption 3.7, we derive dBL(P ◦ [v(θ)]−1, P ◦ [v(0)]−1) = lim k→∞ dBL(P ◦ [vnk (θ)]−1, P ◦ [vnk (0)]−1) = lim k→∞ sup ∥φ∥BL≤1 (∫ l2ρ φdP ◦ [vnk (θ)]−1 − ∫ l2ρ φdP ◦ [vnk (0)]−1 ) = lim k→∞ sup ∥φ∥BL≤1 (∫ Ω φ(vnk (θ))dP − ∫ Ω φ(vnk (0))dP ) EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 11 = lim k→∞ sup ∥φ∥BL≤1 ( 1 nk + 1 nk∑ N=0 ∫ Ω [φ(u((N + 1)θ))− φ(u(Nθ))]dP ) = lim k→∞ 1 nk + 1 nk∑ N=0 dBL(P ◦ [u((N + 1)θ)]−1, P ◦ [u(Nθ)]−1) = 0. That is to say, v(θ) has the same distribution as v(0). We define z(t) : R+ → l2ρ by z(t) = v(t− ntθ), where nt = max{n ∈ N|nθ < t}. Hence, z(t) is a θ-periodic solution in distribution of (2.3). □ 4. Applications It is worth noting that Lyapunov’s method can also be applied to prove the existence of periodic solution in distribution. Example 4.1. Consider the equation of motion of Hooke’s law in l2ρ: dui(t) = [ν(ui+1(t)− 2ui(t) + ui−1(t))− λui(t)]dt+ σi(t, u(t))dWi(t), (4.1) where λ > 0 is a constant. Suppose that Assumption 2.1 holds. Here −λui de- scribes the strength of negative feedback, where λ > 2L. We make the following assumptions: (A1) ρ(i) ≤ c0ρ(i ± 1), for all i ∈ Z, where c0 is a positive constant with c0 < 1 + λ−2L 2ν . (A2) there exists a positive constant c1 such that (2νc0 − 2ν − λ)∥u∥4ρ + ∥u∥2ρ|σ(t, u)|2 + |⟨u, σ(t, u)⟩|2 ≤ −c1∥u∥4ρ for all u ∈ l2ρ, t ∈ R+. (A3) For all u, v ∈ l2ρ, t ∈ R+, there exists a constant c2 such that ∞∑ i=1 JT σi (t, u)Jσi(t, v) ≤ −c2I where c2 > 2(2c0ν − 2ν − λ)/3. Then (4.1) admits a periodic solution in distribution. We define f(t, u) = ∥u∥4ρ. By (A1) and (A2), we have Lf(t, u) = ft(t, u) + ⟨fu(t, u),−νAu− λu⟩+ 1 2 trace(σT (t, u)fuu(t, u)σ(t, u)) ≤ 4ν∥u∥2ρ ∑ i∈Z ρiui(ui+1 − 2ui + ui−1)− 4λ∥u∥4ρ + 2∥u∥2ρ|σ(t, u)|2 + 4|⟨u, σ(t, u)⟩|2 ≤ 4ν∥u∥2ρ ∑ i∈Z ρiui(c0ui − 2ui + c0ui)− 4λ∥u∥4ρ + 2∥u∥2ρ|σ(t, u)|2 + 4|⟨u, σ(t, u)⟩|2 = 4(2νc0 − 2ν − λ)∥u∥4ρ + 2∥u∥2ρ|σ(t, u)|2 + 4|⟨u, σ(t, u)⟩|2 ≤ −4c1∥u∥4ρ. 12 Y. GAO, X. YANG EJDE-2024/25 So, we have E[f(t, uξ(t))] = E[f(0, ξ)] + E ∫ t 0 Lif(s, u(s))ds ≤ E[f(0, ξ)]− 4c1 ∫ t 0 E[f(s, u(s))]ds. It follows from Gronwall’s inequality that for all E∥ξ∥4ρ < ∞, and t ∈ R+, that E∥uξ(t)∥4ρ ≤ E[f(t, uξ(t))] ≤ E[f(0, ξ)]e−4c1t. Therefore, Assumption 3.6 is fulfilled. Furthermore, by (A1), Lf(t, u− v) = ft(t, u− v) + ⟨fu(t, u− v),−νA(u− v)− λ(u− v)⟩ + 1 2 trace[(σ(t, u)− σ(t, v))T fuu(t, u− v)(σ(t, u)− σ(t, v))] = 4ν∥u− v∥2ρ ∑ i∈Z ρi(ui − vi)[(ui+1 − vi+1)− 2(ui − vi) + (ui−1 − vi−1)] − 4λ∥u− v∥4ρ + 6∥u− v∥2ρ(u− v)T [ ∞∑ i=1 (∫ 1 0 JT σi (t, v + s(u− v))ds ) × (∫ 1 0 JT σi (t, v + s(u− v))ds )] (u− v) ≤ 4ν(2c0 − 2)∥u− v∥4ρ − 4λ∥u− v∥4ρ + 6∥u− v∥2ρ(u− v)T [ ∞∑ i=1 (∫ 1 0 JT σi (t, v + s(u− v))ds ) × (∫ 1 0 JT σi (t, v + s(u− v))ds )] (u− v) = [4(2c0ν − 2ν − λ)− 6c2] ∥u− v∥4ρ. Let c3 = 4(2c0ν − 2ν − λ) − 6c2. For any given ξ, η ∈ L2(Ω, l2ρ), applying Itô’s formula to f(t, uξ(t)− uη(t))e −c3t, we obtain E[f(t, uξ(t)− uη(t))e −c3t] = E[f(0, ξ − η)] + ∫ t 0 −c3e −c3sE[f(s, uξ(s)− uη(s))]ds + ∫ t 0 e−c3sE[Lif(s, uξ(s)− uη(s))]ds ≤ E[f(0, ξ − η)]. Hence E[f(t, uξ(t) − uη(t))] ≤ E[f(0, ξ − η)]ec3t for each t ∈ R+. In addition, for each t ∈ R+, there exists k ∈ N such that t ∈ [kθ, kθ + θ]. By Assumption 2.1 and [20, Theorem 1.7.1], we have E∥uξ(t)− uη(t)∥4ρ ≤ 27E∥uξ(kθ)− uη(kθ)∥4ρ + 27t3E ∫ t kθ ∥ − νA(uξ(s)− uη(s))− λ(uξ(s)− uη(s)) + f(uξ(s))− f(uη(s))∥4ρds+ 972TE ∫ t kθ ∥σ(s, uξ(s))− σ(s, uη(s))∥4ρds EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 13 ≤ 27E∥uξ(kθ)− uη(kθ)∥4 + (9T )3E ∫ t kθ ∥ − νA(uξ(s)− uη(s))∥4ρds + (9T )3E ∫ t kθ ∥ − λ(uξ(s)− uη(s))∥4ρds+ (9T )3E ∫ t kθ ∥f(uξ(s))− f(uη(s))∥4ρds + 972TE ∫ t kθ ∥σ(s, uξ(s))− σ(s, uη(s))∥4ρds = 27E∥uξ(kθ)− uη(kθ)∥4 + 486(9T )3ν4 ∫ t kθ E∥uξ(s)− uη(s))∥4ρds + (9T )3λ4 ∫ t kθ E∥uξ(s)− uη(s))∥4ρds+ (9T )3L2 ∫ t kθ E∥uξ(s)− uη(s))∥4ρds + 972TL2 ∫ t kθ E∥uξ(s)− uη(s))∥4ρds = 27E∥uξ(kθ)− uη(kθ)∥4ρ + c4 ∫ t kθ E∥uξ(kθ)− uη(kθ)∥4ρds, where c4 := (9T )3(486ν4 + λ4 + L2}) + 972L2T . Applying Gronwall’s inequality, we have E∥uξ(t)− uη(t)∥4ρ ≤ 27E∥uξ(kθ)− uη(kθ)∥4ρec4T ≤ 27ec4TE[f(kθ, uξ(kθ)− uη(kθ))] ≤ 27ec4T+c3kθE[f(0, ξ − η)] = φ(kθ)E∥ξ − η∥2ρ, (4.2) where φ(t) := 27ec4T+c3kθ. Note that c3 < 0, we have limk→∞ φ(kθ) = 0. Hence, there exists a k0 > 0 such that supk>k0 φ(kθ) < 1. By the contraction mapping fixed point theorem, there exists a unique fixed point ξ∗ ∈ L4(Ω, l4ρ) such that uξ∗(kθ) = ξ∗ for any k > k0. Thus, lim n→∞ 1 n+ 1 n∑ k=0 dBL(P ◦ [uξ∗((k + 1)θ)]−1, P ◦ [uξ∗(kθ)] −1) = lim n→∞ 1 n+ 1 ( k0∑ k=0 dBL(P ◦ [uξ∗((k + 1)θ)]−1, P ◦ [uξ∗(kθ)] −1) + n∑ k=k0 dBL(P ◦ [uξ∗((k + 1)θ)]−1, P ◦ [uξ∗(kθ)] −1) ) = 0 Hence Assumption 3.7 is satisfied. Therefore, (4.1) admits a periodic solution in distribution. Acknowledgments. X. Yang was supported by National Natural Science Founda- tion of China (12071175, 12371191). We extend our heartfelt gratitude to Professor Yong Li for his helpful advice, patient guidance and encouragement. References [1] K. B. Athreya, S. N. Lahiri; Measure theory and probability theory, Springer, New York, (2006). [2] P. W. Bates, K. Lu, B. Wang; Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143-153. 14 Y. GAO, X. YANG EJDE-2024/25 [3] P. W. Bates, H. Lisei, K. Lu; Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. [4] H. Bessaih, M. J. Garrido-Atienza, X. Han, B. Schmalfuss; Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 49 (2017), 1495-1518. [5] F. Chen, Y. Han, Y. Li, X. Yang; Periodic solutions of Fokker-Planck equations, J. Differential Equations, 263 (2017), 285-298. [6] S. N. Chow, J. M. Paret, W. Shen; Traveling waves in lattice dynamical systems, J. Differ- ential Equations, 149 (1998), 248-291. [7] S. N. Ethier, T. G. Kurtz; Markov Processes. Characterization and Convergence, John Wiley, New York, (1986). [8] X. Han, W. Shen, S. Zhou; Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. [9] C. Ji, X. Yang, Y. Li; Periodic solutions for SDEs through upper and lower solutions, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4737-4754. [10] M. Ji, W. Qi, Z. Shen, Y. Yi; Existence of periodic probability solutions to Fokker-Planck equations with applications, J. Funct. Anal., 277 (2019), 41 pp. [11] X. Jiang, Y. Li, X. Yang; LaSalle-type stationary oscillation principle for stochastic affine periodic systems, Stoch. Dyn., 22 (2022), 22 pp. [12] X. Jiang, Y. Li; Wong-Zakai approximations and periodic solutions in distribution of dissi- pative stochastic differential equations, J. Differential Equations, 274 (2021), 652-765. [13] N. I. Karachalios, A. N. Yannacopoulos; Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123. [14] D. Li, Y. Lin, Z. Pu; Non-autonomous stochastic lattice systems with Markovian switching, Discrete Contin. Dyn. Syst., 43 (2023), 1860-1877. [15] D. Li, B. Wang, X. Wang; Periodic measures of stochastic delay lattice systems, J. Differential Equations, 272 (2021), 74-104. [16] Z. Liu, W. Wang; Farvard separation method for almost periodic stochastic differential equa- tions, J. Differential Equations, 260 (2016), 8109-8136. [17] S. Lu, X. Yang; Stability and rate of decay for solutions to stochastic differential equations with Markov switching, Electron. J. Differential Equations, 2024 (2024), no. 01, 1-16. [18] G. Lv, H. Gao, J. Wei; Periodic solution of stochastic process in the distributional sense, J. Evol. Equ., 21 (2021), 4005-4037. [19] X. Mao; Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, (2008). [20] X. Mao, C. Yuan; Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, (2006). [21] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Vol. I, GauthiersVillars, Paris, (1892). [22] H. Poincaré; Les méthodes nouvelles de la mécanique céleste, Vol. II, GauthiersVillars, Paris, (1893). [23] H. Poincaré; Les méthodes nouvelles de la mécanique céleste, Vol. III, GauthierVillars, Paris, (1899). [24] W. Shen; Lifted lattices, hyperbolic structure, and topological disorder in coupled map lat- tices, SIAM J. Appl. Math., 56 (1996), 1379-1399. [25] A. V. Skorokhod; Asymptotic methods in the theory of stochastic differential equations, American Mathematical Society, Providence, RI., (1989). [26] X. Wang, P. E. Kloeden, X. Han; Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise, NoDEA Nonlinear Differnetial Equations Appl., 28 (2021), 31pp. [27] X. Zhou, Y. Li, X. Jiang; Periodic solutions in distribution of stochastic lattice differential equations, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 1300-1322. [28] X. Zhou, J. Xing, X. Jiang, Y. Li; Periodic solutions in distribution of mean-field stochastic differential equations, J. Stat. Phys., 190 (2023), 34 pp. Yue Gao School of Mathematics, Jilin University, Changchun 130012, China Email address: gaoyue19972022@163.com EJDE-2024/25 PERIODIC SOLUTIONS TO SLDES 15 Xue Yang School of Mathematics, Jilin University, Changchun 130012, China Email address: xueyang@jlu.edu.cn 1. Introduction 2. Preliminaries Basic notation Well-posedness of SLDEs 3. Existence of periodic solutions 4. Applications Acknowledgments References